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Finite Group Z3


The unique group of Order 3. It is both Abelian and Cyclic. Examples include the Point Groups $C_3$ and $D_3$ and the integers under addition modulo 3. The elements $A_i$ of the group satisfy ${A_i}^3=1$ where 1 is the Identity Element. The Cycle Graph is shown above, and the Multiplication Table is given below.

$Z_3$ 1 $A$ $B$
1 1 $A$ $B$
$A$ $A$ $B$ 1
$B$ $B$ 1 $A$

The Conjugacy Classes are $\{1\}$, $\{A\}$,

$\displaystyle A^{-1}AA$ $\textstyle =$ $\displaystyle A$  
$\displaystyle B^{-1}AB$ $\textstyle =$ $\displaystyle A,$  

and $\{B\}$,
$\displaystyle A^{-1}BA$ $\textstyle =$ $\displaystyle B$  
$\displaystyle B^{-1}BB$ $\textstyle =$ $\displaystyle B.$  

The irreducible representation (Character Table) is therefore

$\Gamma$ 1 $A$ $B$
$\Gamma_1$ 1 1 1
$\Gamma_2$ 1 1 $-1$
$\Gamma_3$ 1 $-1$ 1

© 1996-9 Eric W. Weisstein